m (Protected "VectorFormulas" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite)))
 
(37 intermediate revisions by 8 users not shown)
Line 1: Line 1:
 +
[[Category:Formulas]]
 +
 +
keywords: vector, gradient, curl, laplacian
 +
 +
<center><font size= 4>
 +
'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
 +
</font size>
 +
 +
'''Vector Identities and Operator Definitions'''
 +
 +
(Used in [[ECE311]])
 +
 +
</center>
 +
 +
----
 +
 
{|
 
{|
−
|-
 
−
! colspan="2" style="background: #e4bc7e; font-size: 110%;" | Vector Identities and Operator Definitions
 
 
|-
 
|-
−
! colspan="2" style="background: #eee;" | Vector Identities
+
! style="background: rgb(228, 188, 126) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Vector Identities and Operator Definitions
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math>\bold{x}\cdot \left(\bold{y}\times \bold{z}\right)= \left(\bold{x}\times \bold{y}\right)\cdot \bold{z}</math>
+
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Vector Identities
−
|-  
+
−
| align="right" style="padding-right: 1em;" | place note here || <math>\bold{x}\times \left(\bold{y}\times \bold{z} \right)=\bold{y}\left(\bold{x} \cdot \bold{z} \right)-\bold{z} \left( \bold{x}\cdot\bold{y}\right) </math>
+
−
|-  
+
−
| align="right" style="padding-right: 1em;" | place note here || <math>\left( \bold{x}\times \bold{y}\right)\cdot \left(\bold{z}\times \bold{w} \right)=\left( \bold{x}\cdot \bold{z}\right) \left(\bold{y} \cdot \bold{w} \right)- \left(\bold{x}\cdot\bold{w} \right) \left( \bold{y}\cdot\bold{z}\right) </math>
+
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math> \nabla \left( f+g \right)= \nabla f+ \nabla g </math>
+
| align="right" style="padding-right: 1em;" | Notes
 +
| Identity
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math> \nabla \left( f g \right)= f \nabla g+ g\nabla f </math>
+
| align="right" style="padding-right: 1em;" |  
 +
|  
 +
<math>\bold{x}\cdot \left(\bold{y}\times \bold{z}\right)= \left(\bold{x}\times \bold{y}\right)\cdot \bold{z}</math>  
 +
 
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math> \nabla \cdot \left(\nabla\times \bold{x} \right)= 0 </math>
+
| align="right" style="padding-right: 1em;" |  
 +
| <math>\bold{x}\times \left(\bold{y}\times \bold{z} \right)=\bold{y}\left(\bold{x} \cdot \bold{z} \right)-\bold{z} \left( \bold{x}\cdot\bold{y}\right) </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math> \nabla \times \nabla \bold{x} = 0 </math>
+
| align="right" style="padding-right: 1em;" |  
 +
| <math>\left( \bold{x}\times \bold{y}\right)\cdot \left(\bold{z}\times \bold{w} \right)=\left( \bold{x}\cdot \bold{z}\right) \left(\bold{y} \cdot \bold{w} \right)- \left(\bold{x}\cdot\bold{w} \right) \left( \bold{y}\cdot\bold{z}\right) </math>
 
|-
 
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\nabla \left( \bold{x}\cdot \bold{y}\right)= \bold{y}\times \left(\nabla\times \bold{x}\right)+  \bold{x} \times \left(\nabla\times \bold{y} \right)+ \left(\bold{y}\cdot\nabla \right)\bold{x} + \left( \bold{x}\cdot\nabla\right) \bold{y} </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \nabla \left( f+g \right)= \nabla f+ \nabla g </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \nabla \left( f g \right)= f \nabla g+ g\nabla f </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \nabla \cdot \left( \bold{x}+\bold{y} \right)= \nabla \cdot \bold{x} + \nabla \cdot \bold{y}  </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \nabla \cdot \left( f \bold{x}\right)= \bold{x} \cdot \nabla f + f \left( \nabla \cdot\bold{x} \right) </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \nabla \times \left( \bold{x} + \bold{y} \right)= \nabla \times \bold{x} + \nabla \times \bold{y} </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \nabla \times \left( u \bold{x} \right)= \left( \nabla u \right) \times \bold{x} + u \left( \nabla \times \bold{x} \right) </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \nabla \cdot \left( \bold{x}\times \bold{y}\right)= \bold{y} \cdot \left( \nabla \times \bold{x}\right) - \bold{x} \cdot \left( \nabla \times \bold{y}\right) </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \nabla \cdot \left(\nabla\times \bold{x} \right)= 0 </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \nabla \times \left( \bold{x} \times \bold{y} \right) = \left( \nabla \cdot \bold{y} \right) \bold{x} - \left( \nabla \cdot \bold{x} \right) \bold{y} + \left( \bold{y} \cdot \nabla \right) \bold{x} - \left( \bold{x} \cdot \nabla \right) \bold{y}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \nabla \times \nabla \bold{x} = 0 </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \nabla ( \bold{C} \cdot \bold{r} ) = \bold{C}  \qquad  \text{where }\bold{C}\text{ is a constant (real or complex)}</math> 
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \nabla \times \left( \nabla \times \bold{x} \right) = \nabla \left( \nabla \cdot \bold{x} \right) - \nabla^2 \bold{x}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \left( \bold{A} \cdot \nabla \right) \bold{B} = \hat{\bold{x}} ( \bold{A}_x \frac{\partial \bold{B}_x}{\partial x} +  \bold{A}_y \frac{\partial \bold{B}_x}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_x}{\partial z} ) + \hat{\bold{y}} ( \bold{A}_x \frac{\partial \bold{B}_y}{\partial x} +  \bold{A}_y \frac{\partial \bold{B}_y}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_y}{\partial z} ) + \hat{\bold{z}} ( \bold{A}_x \frac{\partial \bold{B}_z}{\partial x} +  \bold{A}_y \frac{\partial \bold{B}_z}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_z}{\partial z} )</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{d \left( \bold{x} \cdot \bold{y} \right)}{d\sigma} =\frac{d \bold{y}}{d\sigma}\cdot \bold{x} + \frac{d \bold{x}}{d\sigma}\cdot \bold{y}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac{d \left( \bold{x} \times \bold{y} \right)}{d\sigma} =\frac{d \bold{y}}{d\sigma}\times \bold{x} + \frac{d \bold{x}}{d\sigma}\times \bold{y}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math> \frac {d ( u \bold{v} )}{d \sigma} = \frac {d u}{d \sigma} \bold{v} + u \frac{d \bold{v}}{d \sigma}</math>
 
|}
 
|}
 +
 
{|
 
{|
 
|-
 
|-
−
! colspan="2" style="background: #eee;" | Vector Operators in Rectangular Coordinates
+
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Vector Operators in Rectangular Coordinates
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here || <math>\nabla f(x,y,z) = \mathbf{\hat x} \frac{\partial f}{\partial x}+\mathbf{\hat y}\frac{\partial f}{\partial y}+\mathbf{\hat z} \frac{\partial f}{\partial z}</math>  
+
| align="right" style="padding-right: 1em;" | Notes
−
|-  
+
| Operator
 +
|
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\nabla f(x,y,z) = \mathbf{\hat x} \frac{\partial f}{\partial x}+\mathbf{\hat y}\frac{\partial f}{\partial y}+\mathbf{\hat z} \frac{\partial f}{\partial z}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\nabla \cdot \bold{v} =  \frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+ \frac{\partial v_z}{\partial z}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\nabla \times \bold{v} =  \mathbf{\hat x} \left( \frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z} \right) +
 +
\mathbf{\hat y} \left( \frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x} \right) +
 +
\mathbf{\hat z} \left( \frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y} \right) </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\nabla^2 f = \frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+ \frac{\partial^2 f}{\partial z^2}</math>
 
|}
 
|}
  
 +
<br>
  
 
{|
 
{|
 
|-
 
|-
−
! colspan="2" style="background: #eee;" | Vector Operators in Spherical Coordinates
+
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Vector Operators in Cylindrical Coordinates
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here ||[[Formula_contributed_by_Anshita| <math>\nabla f(\rho,\phi,z) = {\partial f \over \partial \rho}\boldsymbol{\hat \rho}  
+
| align="right" style="padding-right: 1em;" | Notes
 +
| Operator
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\nabla f(\rho,\phi,z) = {\partial f \over \partial \rho}\boldsymbol{\hat \rho}  
 
   + {1 \over \rho}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}  
 
   + {1 \over \rho}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}  
−
   + {\partial f \over \partial z}\boldsymbol{\hat z}</math> ]]
+
   + {\partial f \over \partial z}\boldsymbol{\hat z}</math>  
−
|-  
+
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\nabla \cdot \bold{v} =
 +
\frac{1}{\rho} \frac{\partial \rho v_{\rho}}{\partial \rho}
 +
+
 +
\frac{1}{\rho} \frac{\partial  v_{\phi}}{\partial \phi}
 +
+
 +
\frac{\partial  v_z}{\partial z}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\nabla \times \bold{v} = \boldsymbol{\hat \rho}  ( \frac{1}{\rho} \frac{\partial \bold{v}_z }{\partial \phi} - \frac{\partial \bold{v}_\phi}{\partial z} ) + \boldsymbol{\hat \phi}  ( \frac{\partial \bold{v}_\rho}{\partial z} - \frac{\partial \bold{v}_z}{\partial \rho} ) + \hat{\bold{z}} ( \frac{1}{\rho} \frac{\partial ( \rho \bold{v}_\phi )}{\partial \rho} - \frac{1}{\rho} \frac{\partial \bold{v}_\rho}{\partial \phi} )</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\nabla^2 f =
 +
\frac{1}{\rho} \frac{\partial }{\partial \rho}
 +
\left( \rho  \frac{\partial  f}{\partial \rho}\right)
 +
+
 +
 
 +
\frac{1}{\rho^2} \frac{\partial^2  f}{\partial \phi^2}
 +
+
 +
\frac{\partial^2  f}{\partial z^2}</math>
 
|}
 
|}
  
 +
<br>
  
 
{|
 
{|
 
|-
 
|-
−
! colspan="2" style="background: #eee;" | Vector Operators in Cylindrical Coordinates
+
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Vector Operators in Spherical Coordinates
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | place note here ||[[Formula_contributed_by_Anshita| <math>\nabla f(x,y,z) = {\partial f \over \partial r}\boldsymbol{\hat r}  
+
| align="right" style="padding-right: 1em;" | Notes
 +
| Operator
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\nabla f(x,y,z) = {\partial f \over \partial r}\boldsymbol{\hat r}  
 
   + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta}  
 
   + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta}  
−
   + {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}</math>]]  
+
   + {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}</math>
−
|-  
+
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\nabla \cdot \bold{v} =
 +
\frac{1}{r^2} \frac{\partial r^2 v_r}{\partial r}
 +
+
 +
 
 +
\frac{1}{r\sin\theta} \frac{\partial \sin\theta v_{\theta}}{\partial \theta}
 +
+
 +
\frac{1}{r\sin\theta} \frac{\partial  v_{\phi}}{\partial \phi}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\nabla \times \bold{v} = \frac{\boldsymbol{\hat r } }{r \sin \theta} [ \frac{\partial ( \sin \theta \bold{v}_\phi )}{\partial \theta} - \frac{\partial \bold{v}_\theta}{\partial \phi} ] + \frac { \boldsymbol{\hat \theta} }{r} [ \frac{1}{\sin \theta} \frac{\partial \bold{v}_r}{\partial \phi} - \frac{\partial ( r \bold{v}_\phi )}{\partial r} ] + \frac {\boldsymbol{\hat \phi} }{r} [ \frac{\partial ( r \bold{v}_\theta )}{\partial r} - \frac{\partial \bold{v}_r}{\partial \theta} ]</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" |
 +
| <math>\nabla^2 f =
 +
\frac{1}{r^2} \frac{\partial }{\partial r}
 +
\left( r^2 \frac{\partial  f}{\partial r}\right)
 +
+
 +
 
 +
\frac{1}{r^2 \sin \theta} \frac{\partial }{\partial \theta}
 +
\left(\sin \theta \frac{\partial f}{\partial \theta} \right)
 +
+
 +
\frac{1}{r^2 \sin^2 \theta}\frac{\partial^2  f}{\partial \phi^2}</math>
 
|}
 
|}
 +
<br>
  
 +
{|
 +
|-
 +
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Vector Integral formulas
 +
|-
 +
| align="right" style="padding-right: 1em;" | Notes
 +
| Operator
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\oint_S \bold{A} \cdot d \bold{a} = \int_V \nabla \cdot \bold{A} d \tau \qquad \text{(Divergence therorem)}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\oint_C \bold{A} \cdot d \bold{s} = \int_S ( \nabla \times \bold{A} ) \cdot d \bold{a} \qquad \textrm{(Stokes'  therorem)}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\oint_S u d \bold{a} = \int_V \nabla u d \tau</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\oint_S \bold{A} \times d \bold{a} = - \int_V ( \nabla \times \bold{A} ) d \tau </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\oint_C u d \bold{s} = - \int_S \nabla u \times d \bold{a} </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\oint_S u \bold{A} \cdot d \bold{a} = \int_V [ \bold{A} \cdot ( \nabla u ) + u ( \nabla \cdot \bold{A} ) ]d \tau </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\oint_S \bold{B} ( \bold{A} \cdot d \bold{a} ) = \int_V [( \bold{A} \cdot \nabla ) \bold{B} + \bold{B} ( \nabla \cdot \bold{A}) ] d \tau </math>
 +
|-
 +
 +
|}
 +
<br>
 +
 +
{|
 +
|-
 +
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Formulas Involving Relative Coordinates
 +
|-
 +
| align="right" style="padding-right: 1em;" | Notes
 +
| Operator
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math> \frac{\partial f ( \bold{R} )}{\partial x} = - \frac{\partial f ( \bold{R} )}{\partial x^{'}}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
|<math> \nabla f ( \bold{R} ) = - \nabla^{'} f ( \bold{R} )</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\nabla \cdot \bold{A} ( \bold{R} ) = - \nabla^{'} \cdot \bold{A} ( \bold{R} )</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\nabla \times \bold{A} ( \bold{R} ) = - \nabla^{'} \times \bold{A} ( \bold{R} )</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\nabla^2 f ( \bold{R} )= \nabla^{'2} f ( \bold{R} ) </math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\nabla R = - \nabla^{'} R = \frac{\bold{R}}{R} = \hat{\bold{R}}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\nabla ( \frac{1}{R} ) = - \nabla^{'} ( \frac{1}{R} ) = - \frac{\hat{\bold{R}}}{R^2} = - \frac{\bold{R}}{R^3}</math>
 +
|-
 +
| align="right" style="padding-right: 1em;" | <br>
 +
| <math>\nabla^2 ( \frac{1}{R} )= \nabla^{'2} ( \frac{1}{R} ) = 0 \qquad ( \  R \neq 0 \  )</math>
 +
|-
 +
|}
 
----
 
----
−
[[MegaCollectiveTableTrial1|Back to Collective Table]]
+
[[Collective_Table_of_Formulas|Back to Collective Table]]
−
[[Category:Formulas]]
+

Latest revision as of 11:58, 24 February 2015


keywords: vector, gradient, curl, laplacian

Collective Table of Formulas

Vector Identities and Operator Definitions

(Used in ECE311)

Vector Identities and Operator Definitions
Vector Identities
Notes Identity

$ \bold{x}\cdot \left(\bold{y}\times \bold{z}\right)= \left(\bold{x}\times \bold{y}\right)\cdot \bold{z} $

$ \bold{x}\times \left(\bold{y}\times \bold{z} \right)=\bold{y}\left(\bold{x} \cdot \bold{z} \right)-\bold{z} \left( \bold{x}\cdot\bold{y}\right) $
$ \left( \bold{x}\times \bold{y}\right)\cdot \left(\bold{z}\times \bold{w} \right)=\left( \bold{x}\cdot \bold{z}\right) \left(\bold{y} \cdot \bold{w} \right)- \left(\bold{x}\cdot\bold{w} \right) \left( \bold{y}\cdot\bold{z}\right) $
$ \nabla \left( \bold{x}\cdot \bold{y}\right)= \bold{y}\times \left(\nabla\times \bold{x}\right)+ \bold{x} \times \left(\nabla\times \bold{y} \right)+ \left(\bold{y}\cdot\nabla \right)\bold{x} + \left( \bold{x}\cdot\nabla\right) \bold{y} $
$ \nabla \left( f+g \right)= \nabla f+ \nabla g $
$ \nabla \left( f g \right)= f \nabla g+ g\nabla f $
$ \nabla \cdot \left( \bold{x}+\bold{y} \right)= \nabla \cdot \bold{x} + \nabla \cdot \bold{y} $
$ \nabla \cdot \left( f \bold{x}\right)= \bold{x} \cdot \nabla f + f \left( \nabla \cdot\bold{x} \right) $
$ \nabla \times \left( \bold{x} + \bold{y} \right)= \nabla \times \bold{x} + \nabla \times \bold{y} $
$ \nabla \times \left( u \bold{x} \right)= \left( \nabla u \right) \times \bold{x} + u \left( \nabla \times \bold{x} \right) $
$ \nabla \cdot \left( \bold{x}\times \bold{y}\right)= \bold{y} \cdot \left( \nabla \times \bold{x}\right) - \bold{x} \cdot \left( \nabla \times \bold{y}\right) $
$ \nabla \cdot \left(\nabla\times \bold{x} \right)= 0 $
$ \nabla \times \left( \bold{x} \times \bold{y} \right) = \left( \nabla \cdot \bold{y} \right) \bold{x} - \left( \nabla \cdot \bold{x} \right) \bold{y} + \left( \bold{y} \cdot \nabla \right) \bold{x} - \left( \bold{x} \cdot \nabla \right) \bold{y} $
$ \nabla \times \nabla \bold{x} = 0 $
$ \nabla ( \bold{C} \cdot \bold{r} ) = \bold{C} \qquad \text{where }\bold{C}\text{ is a constant (real or complex)} $
$ \nabla \times \left( \nabla \times \bold{x} \right) = \nabla \left( \nabla \cdot \bold{x} \right) - \nabla^2 \bold{x} $
$ \left( \bold{A} \cdot \nabla \right) \bold{B} = \hat{\bold{x}} ( \bold{A}_x \frac{\partial \bold{B}_x}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_x}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_x}{\partial z} ) + \hat{\bold{y}} ( \bold{A}_x \frac{\partial \bold{B}_y}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_y}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_y}{\partial z} ) + \hat{\bold{z}} ( \bold{A}_x \frac{\partial \bold{B}_z}{\partial x} + \bold{A}_y \frac{\partial \bold{B}_z}{\partial y} + \bold{A}_z \frac{\partial \bold{B}_z}{\partial z} ) $
$ \frac{d \left( \bold{x} \cdot \bold{y} \right)}{d\sigma} =\frac{d \bold{y}}{d\sigma}\cdot \bold{x} + \frac{d \bold{x}}{d\sigma}\cdot \bold{y} $
$ \frac{d \left( \bold{x} \times \bold{y} \right)}{d\sigma} =\frac{d \bold{y}}{d\sigma}\times \bold{x} + \frac{d \bold{x}}{d\sigma}\times \bold{y} $
$ \frac {d ( u \bold{v} )}{d \sigma} = \frac {d u}{d \sigma} \bold{v} + u \frac{d \bold{v}}{d \sigma} $
Vector Operators in Rectangular Coordinates
Notes Operator
$ \nabla f(x,y,z) = \mathbf{\hat x} \frac{\partial f}{\partial x}+\mathbf{\hat y}\frac{\partial f}{\partial y}+\mathbf{\hat z} \frac{\partial f}{\partial z} $
$ \nabla \cdot \bold{v} = \frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+ \frac{\partial v_z}{\partial z} $
$ \nabla \times \bold{v} = \mathbf{\hat x} \left( \frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z} \right) + \mathbf{\hat y} \left( \frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x} \right) + \mathbf{\hat z} \left( \frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y} \right) $

$ \nabla^2 f = \frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+ \frac{\partial^2 f}{\partial z^2} $


Vector Operators in Cylindrical Coordinates
Notes Operator
$ \nabla f(\rho,\phi,z) = {\partial f \over \partial \rho}\boldsymbol{\hat \rho} + {1 \over \rho}{\partial f \over \partial \phi}\boldsymbol{\hat \phi} + {\partial f \over \partial z}\boldsymbol{\hat z} $
$ \nabla \cdot \bold{v} = \frac{1}{\rho} \frac{\partial \rho v_{\rho}}{\partial \rho} + \frac{1}{\rho} \frac{\partial v_{\phi}}{\partial \phi} + \frac{\partial v_z}{\partial z} $
$ \nabla \times \bold{v} = \boldsymbol{\hat \rho} ( \frac{1}{\rho} \frac{\partial \bold{v}_z }{\partial \phi} - \frac{\partial \bold{v}_\phi}{\partial z} ) + \boldsymbol{\hat \phi} ( \frac{\partial \bold{v}_\rho}{\partial z} - \frac{\partial \bold{v}_z}{\partial \rho} ) + \hat{\bold{z}} ( \frac{1}{\rho} \frac{\partial ( \rho \bold{v}_\phi )}{\partial \rho} - \frac{1}{\rho} \frac{\partial \bold{v}_\rho}{\partial \phi} ) $
$ \nabla^2 f = \frac{1}{\rho} \frac{\partial }{\partial \rho} \left( \rho \frac{\partial f}{\partial \rho}\right) + \frac{1}{\rho^2} \frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial z^2} $


Vector Operators in Spherical Coordinates
Notes Operator

$ \nabla f(x,y,z) = {\partial f \over \partial r}\boldsymbol{\hat r} + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta} + {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi} $
$ \nabla \cdot \bold{v} = \frac{1}{r^2} \frac{\partial r^2 v_r}{\partial r} + \frac{1}{r\sin\theta} \frac{\partial \sin\theta v_{\theta}}{\partial \theta} + \frac{1}{r\sin\theta} \frac{\partial v_{\phi}}{\partial \phi} $
$ \nabla \times \bold{v} = \frac{\boldsymbol{\hat r } }{r \sin \theta} [ \frac{\partial ( \sin \theta \bold{v}_\phi )}{\partial \theta} - \frac{\partial \bold{v}_\theta}{\partial \phi} ] + \frac { \boldsymbol{\hat \theta} }{r} [ \frac{1}{\sin \theta} \frac{\partial \bold{v}_r}{\partial \phi} - \frac{\partial ( r \bold{v}_\phi )}{\partial r} ] + \frac {\boldsymbol{\hat \phi} }{r} [ \frac{\partial ( r \bold{v}_\theta )}{\partial r} - \frac{\partial \bold{v}_r}{\partial \theta} ] $
$ \nabla^2 f = \frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2 \frac{\partial f}{\partial r}\right) + \frac{1}{r^2 \sin \theta} \frac{\partial }{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 f}{\partial \phi^2} $


Vector Integral formulas
Notes Operator

$ \oint_S \bold{A} \cdot d \bold{a} = \int_V \nabla \cdot \bold{A} d \tau \qquad \text{(Divergence therorem)} $

$ \oint_C \bold{A} \cdot d \bold{s} = \int_S ( \nabla \times \bold{A} ) \cdot d \bold{a} \qquad \textrm{(Stokes' therorem)} $

$ \oint_S u d \bold{a} = \int_V \nabla u d \tau $

$ \oint_S \bold{A} \times d \bold{a} = - \int_V ( \nabla \times \bold{A} ) d \tau $

$ \oint_C u d \bold{s} = - \int_S \nabla u \times d \bold{a} $

$ \oint_S u \bold{A} \cdot d \bold{a} = \int_V [ \bold{A} \cdot ( \nabla u ) + u ( \nabla \cdot \bold{A} ) ]d \tau $

$ \oint_S \bold{B} ( \bold{A} \cdot d \bold{a} ) = \int_V [( \bold{A} \cdot \nabla ) \bold{B} + \bold{B} ( \nabla \cdot \bold{A}) ] d \tau $


Formulas Involving Relative Coordinates
Notes Operator

$ \frac{\partial f ( \bold{R} )}{\partial x} = - \frac{\partial f ( \bold{R} )}{\partial x^{'}} $

$ \nabla f ( \bold{R} ) = - \nabla^{'} f ( \bold{R} ) $

$ \nabla \cdot \bold{A} ( \bold{R} ) = - \nabla^{'} \cdot \bold{A} ( \bold{R} ) $

$ \nabla \times \bold{A} ( \bold{R} ) = - \nabla^{'} \times \bold{A} ( \bold{R} ) $

$ \nabla^2 f ( \bold{R} )= \nabla^{'2} f ( \bold{R} ) $

$ \nabla R = - \nabla^{'} R = \frac{\bold{R}}{R} = \hat{\bold{R}} $

$ \nabla ( \frac{1}{R} ) = - \nabla^{'} ( \frac{1}{R} ) = - \frac{\hat{\bold{R}}}{R^2} = - \frac{\bold{R}}{R^3} $

$ \nabla^2 ( \frac{1}{R} )= \nabla^{'2} ( \frac{1}{R} ) = 0 \qquad ( \ R \neq 0 \ ) $

Back to Collective Table

Retrieved from "https://www.projectrhea.org/rhea/index.php?title=VectorFormulas&oldid=69156"
Shortcuts
Help Main Wiki Page Random Page Special Pages Log in
Search

Alumni Liaison

Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

Buyue Zhang